Optimal. Leaf size=659 \[ \frac{e^{3/2} \sqrt{c+d x^2} \left (15 a^2 d^2 f-5 a b d (5 c f+3 d e)+3 b^2 c (3 c f+8 d e)\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (15 a^2 d^2 f^2-20 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+9 c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b^3 d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f^2 x \sqrt{c+d x^2} (b c-a d)^2}{b^3 d \sqrt{e+f x^2}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{2 f x \sqrt{c+d x^2} (b c-a d) (2 d e-c f)}{3 b^2 d \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )}{15 b d \sqrt{e+f x^2}}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b} \]
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Rubi [A] time = 0.752705, antiderivative size = 784, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {545, 416, 528, 531, 418, 492, 411, 543, 539} \[ \frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) (b e-a f)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{f x \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e)}{3 b^3 d \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (5 b e-3 a f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 d \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )}{15 b d \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (9 d e-c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b} \]
Antiderivative was successfully verified.
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Rule 545
Rule 416
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rule 543
Rule 539
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx &=\frac{d \int \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2} \, dx}{b}+\frac{(b c-a d) \int \frac{\sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx}{b}\\ &=\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{\int \frac{\sqrt{c+d x^2} \left (e (5 d e-c f)+2 f (3 d e-c f) x^2\right )}{\sqrt{e+f x^2}} \, dx}{5 b}+\frac{((b c-a d) f) \int \frac{\sqrt{c+d x^2} \left (2 b e-a f+b f x^2\right )}{\sqrt{e+f x^2}} \, dx}{b^3}+\frac{\left ((b c-a d) (b e-a f)^2\right ) \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{b^3}\\ &=\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{(b c-a d) \int \frac{c f (5 b e-3 a f)+f (4 b d e+b c f-3 a d f) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^3}+\frac{\int \frac{c e f (9 d e-c f)+f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 b f}\\ &=\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{(c (b c-a d) f (5 b e-3 a f)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^3}+\frac{(c e (9 d e-c f)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 b}+\frac{((b c-a d) f (4 b d e+b c f-3 a d f)) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^3}+\frac{\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 b}\\ &=\frac{(b c-a d) f (4 b d e+b c f-3 a d f) x \sqrt{c+d x^2}}{3 b^3 d \sqrt{e+f x^2}}+\frac{\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt{c+d x^2}}{15 b d \sqrt{e+f x^2}}+\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{(b c-a d) \sqrt{e} \sqrt{f} (5 b e-3 a f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{e^{3/2} (9 d e-c f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{((b c-a d) e f (4 b d e+b c f-3 a d f)) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b^3 d}-\frac{\left (e \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )\right ) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 b d}\\ &=\frac{(b c-a d) f (4 b d e+b c f-3 a d f) x \sqrt{c+d x^2}}{3 b^3 d \sqrt{e+f x^2}}+\frac{\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt{c+d x^2}}{15 b d \sqrt{e+f x^2}}+\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}-\frac{(b c-a d) \sqrt{e} \sqrt{f} (4 b d e+b c f-3 a d f) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 d \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{\sqrt{e} \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b d \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(b c-a d) \sqrt{e} \sqrt{f} (5 b e-3 a f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{e^{3/2} (9 d e-c f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}\\ \end{align*}
Mathematica [C] time = 2.49514, size = 445, normalized size = 0.68 \[ \frac{-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (15 a^2 b d f^2 (2 c f+d e)-15 a^3 d^2 f^3+5 a b^2 f \left (-3 c^2 f^2-7 c d e f+d^2 e^2\right )-3 b^3 e \left (-7 c^2 f^2+c d e f+d^2 e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-i a b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (15 a^2 d^2 f^2-20 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+9 c d e f+d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f \left (a b^2 x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (3 b \left (2 c f+2 d e+d f x^2\right )-5 a d f\right )-15 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 a b^4 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 1939, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}}{a + b x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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